COMMUNICATIOS TIIEORV OF SFAKKCV SYSTE.\[S 699 



insure that it be statistically reasonable. These consistency conditions pro- 

 duce corresponding consistency conditions in the cryptogram. The key gives 

 a certain amount of freedom to the cryptogram but, as more and more 

 letters are intercepted, the consistency conditions use up the freedom al- 

 lowed by the key. Eventually there is only one message and key which 

 satishes all the conditions and we have a unique solution. In the random 

 cipher the consistency conditions are, in a sense "orthogonal" to the "grain 

 of the key" and have their full effect in eliminating messages and keys as 

 rapidly as possible. This is the usual case. However, by proper design it 

 is possible to "line up" the redundancy of the language with the "grain of 

 the key" in such a way that the consistency conditions are automatically 

 satisfied and IIe(K) does not approach zero. These "ideal" systems, which 

 will be considered in the next section, are of such a nature that the trans- 

 formations Ti all induce the same probabilities in the E space. 



17. Ideal Secrecy Systems, 



We have seen that perfect secrecy requires an infinite amount of key if 

 we allow messages of unlimited length. With a finite key size, the equivoca- 

 tion of key and message generally approaches zero, but not necessarily so. 

 In fact it is possible for He(K) to remain constant at its initial value H{K). 

 Then, no matter how much material is intercepted, there is not a unique 

 solution but many of comparable probability. We will define an "ideal" 

 system as one in which He(K) and He(M) do not approach zero as .V — ^ co , 

 A "strongly ideal" system is one in which He{K) remains constant 

 at H{K). 



An example is a simple substitution on an artificial language in which 

 all letters are equiprobable and successive letters independently chosen. 

 It is easily seen that He(K) — H(K) and He(M) rises linearly along a line 

 of slope log G (w^here G is the number of letters in the alphabet) until it 

 strikes the line H{K), after which it remains constant at this value. 



With natural languages it is in general possible to approximate the ideal 

 characteristic — the unicity point can be made to occur for as large A as is 

 desired. The complexity of the system needed usually goes up rapidly when 

 we attempt to do this, however. It is not always possible to attain actually 

 the ideal characteristic with any system of finite complexity. 



To approximate the ideal equiv^ocation, one may first operate on the 

 message with a transducer which removes all redundancies. After this almost 

 any simple ciphering system — substitution, transposition, \'igenere, etc., 

 is satisfactory. The more elaborate the transducer and the nearer the 

 output is to the desired form, the more closely will the secrecy system ap- 

 proximate the ideal characteristic. 



